Edge irredundant colorings in graphs

An edge irredundant coloring of a graph G = (V, E) is an edge partition ∏={E1,E2,…,Ek} of E into nonempty edge irredundant sets. The edge irratic number is the minimum order of an edge irredundant coloring of G and it is denoted by χir′(G). In this paper a study has been initiated on edge irredundant coloring of G. We have characterized all graphs G for which χir′(G)=m or m−1, where m is the size of a graph G. Also we have obtained some bounds on χir′ for triangle-free graphs and trees.